- #1
PatsyTy
- 30
- 1
Homework Statement
Prove that ##[L_i,x_j]=i\hbar \epsilon_{ijk}x_k \quad (i, j, k = 1, 2, 3)## where ##L_1=L_x##, ##L_2=L_y## and ##L_3=L_z## and ##x_1=x##, ##x_2=y## and ##x_3=z##.
Homework Equations
There aren't any given except those in the problem, however I assume we use
$$L_i=x_jp_k-p_kx_j$$
$$[A,B]=AB-BA$$
$$[A+B,C]=[A,C]+[B,C]$$
$$[AB,C]=[A,C]B+A[B,C]$$
The Attempt at a Solution
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Sub ##L_i=x_jp_k-p_kx_j## into ##[L_i,x_j]##
$$[x_j-p_k-p_kx_j]=[x_jp_k,x_j]-[p_kx_j,x_j]$$
Use ##[AB,C]=[A,C]B+A[B,C]##
$$([x_j,x_j]p_k+x_j[p_k,x_j])-([p_k,x_j]x_j+[x_j,x_j]p_k)$$
The two commutators ##[x_j,x_j]=0## so I end up with
$$x_j[p_k,x_j]-[p_k,x_j]x_j$$
However I have no idea where to go from here. I don't understand where the ##x_k## term comes from. I assume the ##i\hbar## term would appear if I wrote the momentum operators as ##-i\hbar\frac{\partial}{\partial x_k}## however then I would end up with a bunch of differential operators I don't know how to work with.